For integer w > 2, x^w + y^w = z^w, where x y z are integers, has no positive integer solutions {Fermat's last theorem} {Fermat last theorem}. Fermat proved that x^3 + y^3 = z^3 has no positive integer solutions. Andrew Wiles proved theorem for all cases [1993].
properties
x^2 + y^2 = z^2 has solutions for x = 3, 4, 5, and so on. z is odd, x is odd, and y is even. x + y > z. In parameterization, x = 2*m*n, y = m^2 - n^2, and z = m^2 + n^2.
Multiples of Pythagorean triples are Pythagorean triples. For lowest Pythagorean triples, z - y = 1 if x is odd, or z - y = 2 if x is even. For lowest Pythagorean triples, n = 1, if lowest of x or y is even, or m - n = 1, if lowest of x or y is odd.
x^2 + y^2 = z^2 has (x^2)/(z^2) + (y^2)/(z^2) = 1 and ((x^2 + y^2)^0.5)/z = 1, so one percent plus other percent = 100%. x^2/z^2 + y^2/z^2 + 2*x*y/z^2 = 1 + 2*x*y/z^2 means (x + y)^2 = z^2 + 2*x*y, where (x + y)^2 is area of square whose side is straight line of x + y, and 2*x*y is two times area of triangle rectangle.
triangle
For x + y = z, three natural numbers lie on a straight line, with xy angle 180 degrees. For x^2 + y^2 = z^2, three natural numbers lie on a right triangle, with xy angle 90 degrees. Perhaps, for x^3 + y^3 = z^3, three natural numbers lie on a triangle with xy angle 60 degrees, but this only allows x = y = z, so no natural number solutions. Perhaps, for x^4 + y^4 = z^4, three natural numbers lie on a triangle with xy angle 45 degrees, but this only allows x < x and z > y, so no natural number solutions.
cube
x^3 + y^3 = z^3 makes (x + y)^3 = z^3 + 3*x^2*y + 3*x*y^2. 3 * x^2 * y + 3 * x * y^2 = 3 * x * y * (x + y). (x + y)^3 is volume of cube with side x + y. 3 * x * y * (x + y) is three times volume of rectilinear solid with sides x, y, and x + y. x + y > z. Perhaps, side lengths and angles make an impossible figure. Perhaps, all higher powers make impossible figures. Perhaps, x^3 + y^3 = z^3 requires not odd and even properties, but three-part system with divisible-by-3 numbers, numbers one higher, and numbers two higher. x, y, and z must come from different categories. Perhaps, this is impossible.
Mathematical Sciences>Number Theory>Theorem
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Date Modified: 2022.0224